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An improved design for hollow antiresonant fibers (HAFs) is presented. It consists of adding extra antiresonant glass elements within the air cladding region of an antiresonant hollow-core fiber. We use numerical simulations to compare fiber structures with and without the additional cladding elements in the near- and mid-IR regimes. We show that realizable fiber structures can provide greatly improved performance in terms of leakage and bending losses compared to previously reported antiresonant fibers. At mid-IR wavelengths, the adoption of this novel fiber design will lead to HAFs with reduced bending losses. In the near-IR, this design could lead to the fabrication of HAFs with very low attenuation.